Volume 31, Issue 1

In this paper, the rate of convergence to the exponential distribution in the limit theorem for extreme values of the queue length in queuing systems M / G / 1 {M/G/1} and G / M / 1 {G/M/1} are studied. In proving these results, we first prove one general boundary theorem for extremal values of regenerative processes, which is of independent interest. This topic is closely related to the task of the first passage times to rare sets by the regenerative process. The need for such research arises in many areas: the mathematical theory of reliability, queuing theory, some statistical problems of physics.

Let R be a ring and let n ≥ 1 {n\geq 1} be a fixed integer. An additive mapping h of a ring R into itself is called an n -Jordan homoderivation if h ⁢ ( x n ) = ( h ⁢ ( x ) + x ) n - x n {h(x^{n})=(h(x)+x)^{n}-x^{n}} holds for all x ∈ R {x\in R} . In this paper, we initiate the study of n -Jordan homoderivations on rings. Precisely, we characterize n -Jordan homoderivations in terms of homoderivations and anti-homomorphisms under certain conditions. Finally, we conclude our paper with a direction for further research.

In this paper, the generalized inverses and the solution of the generalized inverse equation in a specific set are used to characterize the SEP element in a ring with involution.

In this paper, we aim to introduce new inequalities of the m -tuples of semi-Hilbert space operators. These inequalities, which are based on some numerical radius inequalities due to Dragomir et al. for a single operator on a Hilbert space. Further, we give a new formula of the A -Euclidean operator semi-norm of m -tuples of operators on a semi-Hilbert space.

In this paper, we study the existence and multiplicity of solutions for the Schrödinger–Bopp–Podolsky system { - Δ ⁢ u + V ⁢ ( x ) ⁢ u + ϕ ⁢ u = f ⁢ ( u ) + λ ⁢ | u | 4 ⁢ u in  ⁢ ℝ 3 , - Δ ⁢ ϕ + a 2 ⁢ Δ 2 ⁢ ϕ = 4 ⁢ π ⁢ u 2 in  ⁢ ℝ 3 , \left\{\begin{aligned} \displaystyle{-}\Delta u+V(x)u+\phi u&\displaystyle=f(u% )+\lambda|u|^{4}u&&\displaystyle\phantom{}\text{in }\mathbb{R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\text{in }\mathbb{R}^{3},\end{aligned}\right. where x ∈ ℝ 3 {x\in\mathbb{R}^{3}} , a > 0 {a>0} , V ⁢ ( x ) ∈ 𝒞 ⁢ ( ℝ 3 , ℝ ) {V(x)\in\mathcal{C}(\mathbb{R}^{3},\mathbb{R})} . Using variational methods and the symmetric mountain pass theorem, we establish the existence of multiple solutions for this system.

Let ℛ {\mathscr{R}} be a prime ring, which is not commutative, with involution * {*} and symmetric ring of quotients 𝒬 s {\mathscr{Q}_{s}} . The aim of the present paper is to describe the structures of a pair of generalized Jordan * {*} -derivations of prime * {*} -rings. Notably, we prove that if a noncommutative prime ring ℛ {\mathscr{R}} with involution * {*} admits a couple of generalized Jordan derivations ℱ 1 {\mathcal{F}_{1}} and ℱ 2 {\mathcal{F}_{2}} associated with Jordan * {*} -derivations 𝒟 1 {\mathscr{D}_{1}} and 𝒟 2 {\mathscr{D}_{2}} such that ℱ 1 ⁢ ( x ) ⁢ x * - x ⁢ ℱ 2 ⁢ ( x ) = 0 {\mathcal{F}_{1}(x)x^{*}-x\mathcal{F}_{2}(x)=0} for all x ∈ ℛ {x\in\mathscr{R}} , then the following holds: (i) if 𝒟 1 ⁢ ( x ) = 𝒟 2 ⁢ ( x ) {\mathscr{D}_{1}(x)=\mathscr{D}_{2}(x)} , then ℱ 1 ⁢ ( x ) = ℱ 2 ⁢ ( x ) = 0 {\mathcal{F}_{1}(x)=\mathcal{F}_{2}(x)=0} for all x ∈ ℛ {x\in\mathscr{R}} , (ii) if 𝒟 1 ⁢ ( x ) ≠ 𝒟 2 ⁢ ( x ) {\mathscr{D}_{1}(x)\neq\mathscr{D}_{2}(x)} , then there exists q ∈ 𝒬 s {q\in\mathscr{Q}_{s}} such that ℱ 1 ⁢ ( x ) = x ⁢ q {\mathcal{F}_{1}(x)=xq} , and ℱ 2 ⁢ ( x ) = q ⁢ x * {\mathcal{F}_{2}(x)=qx^{*}} for all x ∈ ℛ {x\in\mathscr{R}} . Moreover, some related results are also discussed.

Under GCH , there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the card ⁢ ( G ) {{\rm card}(G)} -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G . Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group 𝐒 1 {{\bf S}_{1}} .

For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established.

Let 𝐴 and 𝐵 be rings and 𝑈 a ( B , A ) (B,A) -bimodule. The Gorenstein projective, injective and flat complexes over the formal triangular matrix ring T = ( A 0 U B ) T=\bigl{(}\!\begin{smallmatrix}A&0\\ U&B\end{smallmatrix}\!\bigr{)} are explicitly described. These statements extend some earlier results of Enochs, Cortés-Izurdiaga, Torrecillas, Zhang and Mao in this direction.

This work deals with a von Kármán system with internal damping. For the solution’s existence, we use nonlinear semigroup theory tools. We construct an evolution system by nonlinear Lipschitz perturbation of a semigroup of contractions. We apply the energy method for the asymptotic behavior, which uses suitable multipliers to construct a Lyapunov functional that leads to exponential decay.

Let G be a finite non-abelian group. The commuting conjugacy class graph Γ ⁢ ( G ) {\Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ ⁢ ( G ) {\Gamma(G)} are connected by an edge if there exist elements x ∈ X {x\in X} and y ∈ Y {y\in Y} such that x ⁢ y = y ⁢ x {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that G Z ⁢ ( G ) {\frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or p 2 ⁢ q {p^{2}q} , is determined.

We consider a generalization of an extremal problem known as the lifeguard problem. In particular, we give a geometric characterization of the optimal path and an algorithm for computing the minimal rescue time.

Let 𝒜 {\mathcal{A}} be a prime ring equipped with an involution ‘ * {*} ’ of order 2 and let m ≠ n {m\neq n} be some fixed positive integers such that 𝒜 {\mathcal{A}} is 2 ⁢ m ⁢ n ⁢ ( m + n ) ⁢ | m - n | {2mn(m+n)|m-n|} -torsion free. Let 𝒬 m ⁢ s ⁢ ( 𝒜 ) {\mathcal{Q}_{ms}(\mathcal{A})} be the maximal symmetric ring of quotients of 𝒜 {\mathcal{A}} and consider the mappings ℱ {\mathcal{F}} and 𝒢 : 𝒜 → 𝒬 m ⁢ s ⁢ ( 𝒜 ) {\mathcal{G}:\mathcal{A}\to\mathcal{Q}_{ms}(\mathcal{A})} satisfying the relations ( m + n ) ⁢ ℱ ⁢ ( a 2 ) = 2 ⁢ m ⁢ ℱ ⁢ ( a ) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ ( a ) (m+n)\mathcal{F}(a^{2})=2m\mathcal{F}(a)a^{*}+2na\mathcal{F}(a) and ( m + n ) ⁢ 𝒢 ⁢ ( a 2 ) = 2 ⁢ m ⁢ 𝒢 ⁢ ( a ) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ ( a ) (m+n)\mathcal{G}(a^{2})=2m\mathcal{G}(a)a^{*}+2na\mathcal{F}(a) for all a ∈ 𝒜 {a\in\mathcal{A}} . Using the theory of functional identities and the structure of involutions on matrix algebras, we prove that if ℱ {\mathcal{F}} and 𝒢 {\mathcal{G}} are additive, then 𝒢 = 0 {\mathcal{G}=0} . We also show that, in case ‘ * * ’ is any nonidentity anti-automorphism, the same conclusion holds if either ‘ * {*} ’ is not identity on 𝒵 ⁢ ( 𝒜 ) {\mathcal{Z}(\mathcal{A})} or 𝒜 {\mathcal{A}} is a PI-ring.

In the present paper, the unique solvability of the direct and inverse source problems for the pseudo-parabolic equation involving the bi-ordinal Hilfer fractional derivative and 2D Landau Hamiltonian operator is considered. Applying the Fourier analysis for the operator Landau Hamiltonian, the theorems of uniqueness and existence of solutions to direct and inverse source problems are proved. In the investigation of the inverse source problem, we have used the value of unknown function at the final time in order to find the right-hand side of the equation. It is also presented the stability result of the inverse problem.

We study the existence of solutions for the nonlinear elliptic problem - Δ ⁢ v - λ ⁢ g 1 ⁢ v + h 1 ⁢ ( v ) = f 1 \displaystyle-\Delta v-\lambda g_{1}v+h_{1}(v)=f_{1} in  ⁢ V ∖ V 0 , \displaystyle\phantom{}\text{in }V\setminus V_{0}, v = 0 \displaystyle v=0 on  ⁢ V 0 , \displaystyle\phantom{}\text{on }V_{0}, where V is the Sierpiński gasket in ℝ N - 1 {\mathbb{R}^{N-1}} ( N ≥ 2 {N\geq 2} ), V 0 {V_{0}} is its boundary (consisting of its N corners) and λ ∈ ℝ {\lambda\in\mathbb{R}} . Here f 1 , g 1 : V → ℝ {f_{1},g_{1}:V\to\mathbb{R}} , h 1 : ℝ → ℝ {h_{1}:\mathbb{R}\to\mathbb{R}} are the maps satisfying suitable hypotheses.